Optimizing Space Robot Configurations to Minimize Capture Contact Forces

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Introduction The Earth’s orbit is becoming dangerously congested. With thousands of active satellites now sharing space with retired spacecraft and millions of debris fragments, the risk of collisions is increasing at an alarming pace [1; 2]. To maintain orbital operations sustainable, robotic capture and servicing technologies have emerged as vital tools for debris removal, satellite life extension, and in-orbit repair [3;4]. Free-floating robotic manipulators, in particular, offer unique capabilities for these missions. However, a serious challenge remains: when a robot arm makes contact with a satellite or debris, the resulting forces can damage sensitive components or destabilize the servicing spacecraft itself [5; 6]. Therefore, reducing the capture contact forces is essential for safe and reliable operation. Various approaches have been explored to address this issue. Simple spring-damper models provided an early approximation of contact but failed to capture nonlinear energy dissipation. Ad-vanced formulations such as Hunt - Crossley [7] and Lankarani - Nikravesh [8] incorporated hyste-resis damping and offered more realistic force-deformation predictions. Alternative models, in-cluding Lee - Wang [9], Flores et al. [10], Gonthier et al. [11], and Hu - Guo [12], further extended the modeling landscape. In parallel, control methods have evolved: sliding mode control [5], impedance and admittance control [13; 14], adaptive control [15], and prescribed performance control [16] were all developed to regulate manipulator behavior. Hybrid force/motion strategies [17] and trajectory optimization approaches [18; 19] further sophisti-cation, whereas reinforcement learning [20; 21; 22] provided data-driven adaptability under uncertainty. Despite this progress, limitations persist. Many studies assume rigid targets [23; 24] and single-point contacts [14], ignoring flexible appendages and multi-contact realities in orbit. Others emphasize trajectory planning [25; 26] but underplay the influence of manipulator configuration. Studies on effective mass [27; 28] confirmed that a robot’s apparent inertia directly affects impact severity, and singularity analyses [29; 30] highlighted the risks of unstable postures. However, few efforts have integrated these insights into a configuration optimization framework. The contribution of this study is the development of a configuration optimization framework that uses integrated effective mass (IEM) as a guiding metric to minimize the capture contact forces. By combining nonlinear Hunt - Crossley modeling, Jacobian-based IEM analysis, and evaluation of manipulator configurations, this study demonstrates that contact forces can be reduced by nearly an order of magnitude while avoiding sin-gularities. This contribution shifts the configuration from a background parameter to a central design tool for safer on-orbit servicing and debris removal. 1. Literature Review The problem of safe capture in space robotics has driven extensivere-searchh across modeling, control, and optimization. Early studies often relied on simple spring-damper systems [31], which offered basic insights but failed to capture the nonlinearities inherent in contact. To improve the realism, Hunt and Crossley [7] and Lankarani and Nikravesh [8] introduced nonlinear damping models that incorporated hysteresis and better reflected energy dissipation. Comparative studies by Flores et al. [32], Gonthier et al. [11], and Hu - Guo [12] confirmed the advantages of these models, with Hunt - Crossley proving especially effective. Control methods were another major line ofre-searchh. Sliding mode control [34] became popular because of its robustness against uncertainties, whereas impedance [35] and admittance control [36] enforced compliance at the manipulator - target interface. Adaptive schemes [37] and prescribed performance control [38] offered resilience against modeling errors, and hybrid force/motion frameworks [39] sought to unify trajectory tracking with force regulation. More recent approaches integrated learning with reinforcement learning methods [40], allowing robots to adapt to unstructured environments. Optimization-based strategies have complemented these advances. Trajectory optimization [41; 42] produced smoother motions that reduced peak impact loads. Particle swarm optimization [43; 44] and deep learning methods [45] expanded the toolkit for path planning under uncertainty. Simultaneously, configuration optimization [46; 47] emerged as a promising method, emphasizing that the manipulator posture itself is a determinant of the contact force. This idea is closely linked to the concept of effective mass [46], which quantifies the apparent inertia of a manipulator in the direction of contact. Research on singularity analysis [48; 49] further highlighted how poor configurations can destabilize capture operations. Despite this breadth ofre-searchh, significant gaps remain. Many models assume rigid targets [37; 46] and simplified contact conditions [50] whereas real-world missions involve flexible structures and multi-contact dynamics. Learning-based controllers [51-53] have demonstrated adaptability but typically prioritize motion planning over posture optimization. Consequently, the explicit use of the integrated effective mass as a central metric for configuration optimization remains underexplored. This study addresses this gap by presenting a unified framework that com-bines nonlinear Hunt - Crossley modeling, Jacobian-based IEM analysis, and systematic configuration optimization to reduce capture forces in space robotics. 2. Methodology Understanding the behavior of contact forces during robotic interactions in space is essential for the safe and effective performance of manipulators in missions such as satellite servicing and debris removal. This study employed a physics-based modeling and simulation approach to estimate, minimize, and regulate these forces under microgravity conditions. The 7-DOF KUKA LBR R800 manipulator [54] (Figure 1), with its 800 mm maximum reach, was adopted as the reference system, as this reach defined the manipulator’s operational workspace and influenced the capture dynamics. The methodology focuses on determining the integrated effective mass (IEM) and related dynamic parameters, providing a foundation for configuration optimization and impact force reduction during on-orbit capture operations. Figure 1. The 7-DOF KUKA LBR iiwa R800 manipulator S o u r c e: by KUKA Roboter GmbH. LBR iiwa - product specification sheet, version v5. Technical datasheet. Augsburg, Germany, 2015. Available from: https://www.kuka.com/en-de/products/robot-systems/industrial-robots/lbr-iiwa (accessed: 12.05.2024) 2.1. Contact Force Modeling In orbital environments where gravity is neg-ligible, the interaction of a robotic manipulator with a target object, such as a tumbling satellite, results in complex contact dynamics. These inter-actions are governed by parameters such as the relative velocity, stiffness of the contact interface, damping, and, crucially, the apparent or effective mass along the direction of contact. Traditional linear models are often insufficient for realistically modeling these forces. Instead, nonlinear formulations, such as the Hunt - Crossley model, are preferred because they incorporate hys-teresis damping, a mechanism that better captures energy dissipation during impact. This allowed for a more accurate estimation of both the peak forces and deformation during contact. Mathematically, the Hunt - Crossley model represents the contact force F as [32; 56-58] where and For single-point and no-friction assumptions, the classical model of the contact force, which incorporates a spring and damper in parallel connecting the contact points, is [58]: where is the stiffness parameter, represent the deformation and deformation velocity; is the nonlinear power exponent, which is considered to be 1.5 in most cases, and is called the hysteresis damping factor with several classical expressions shown in Table 1. This model, along with the variants proposed by Lee and Wang [9], Gonthier et al. [11], and Flores et al. [10], allows impact events to be simulated with greater accuracy. However, among all the compared formulations, the Hunt - Crossley model offers the most reliable balance between computational simplicity and physical realism, particularly in returning to a zero-force state post-deformation (i.e., capturing elastic recovery accurately) [58]. Table 1 Classical Expressions of Hysteresis Damping Factor Model Hysteresis Damping Factor Model Hysteresis Damping Factor Herbert - McWhannell Hunt - Crossley Lankarain - Nikravesh Lee - Wang Flores et al. [10] Gonthier et al. [11] Zhiying - Qishao Hu - Guo S o u r c e: by P. Flores and H.M. Lankarani [10] The maximum deformation occurs when the final velocity is zero, thus for maximum deformation between the manipulator and the target, the maximum force of contact can be determined by 6 N: 2.2. Hysteresis Damping Factor Comparison A critical aspect of realistic force modeling is the selection of the damping factor, λ, which dictates how energy is dissipated. As seen in comparative simulations, while models like Lankarani - Nikravesh [8] and Lee - Wang [9] exhibit good dissipation, the Hunt - Crossley [7] model minimizes estimation error for maximum contact force and better reflects post-impact behavior. To choose the best model to determine the hysteresis damping factor, which is a critical parameter for showcasing the maximum contact force created between the tumbling target and the robotic manipulator, the relationship between the deformation and contact force was modeled as shown in Figure 1. The figure shows the curve of the contact force with respect to deformation, where and and the different models of the hysteresis damping factor are adopted. Figure 2 was plotted using the equations from [10] and simulated in MATLAB to observe the performance of each model. Based on these simulations, among the various numerical models analyzed, the Hunt - Crossley model demonstrated the most promising performance in estimating the maximum contact force with minimal error between the tumbling target and the robotic manipulator. Additionally, its force-displacement loop exhibited a tendency to return to zero more effectively than the other models. a b c d Figure 2. Curve of contact force with respect to deformation for different models of hysteresis damping factor: a - Herbert - McWhannell Model; b - Lankarain - Nikravesh Model; c - Flores et al. Model; d - Zhiying - Qishao Model; e - Hunt - Crossley Model; f - Lee - Wang Model; g - Gonthier et al. Model; h - Hu - Guo Model S o u r c e: by P. Flores and H. M. Lankarani [10] e f g h Figure 2 (Ending). Curve of contact force with respect to deformation for different models of hysteresis damping factor: a - Herbert - McWhannell Model; b - Lankarain - Nikravesh Model; c - Flores et al. Model; d - Zhiying - Qishao Model; e - Hunt - Crossley Model; f - Lee - Wang Model; g - Gonthier et al. Model; h - Hu - Guo Model S o u r c e: by P. Flores and H. M. Lankarani [10] 2.3. Coefficient of Restitution (CR) Another key parameter is the coefficient of restitution, which is defined as the ratio of the post-impact to pre-impact relative velocities. It ef-fectively measures how “bouncy” a collision is and thus determines the extent of rebound. A higher CR (e.g., 0.9) implies more energy retention (greater rebound), whereas a lower CR (e.g., 0.7) reduces the rebound but increases the energy dissipation. In space applications, a balanced CR of approximately 0.8 is optimal, as it minimizes rebound without causing excessive deformation or prolonging the contact duration, which could destabilize the robot or the captured object [58; 59]. 2.4. Integrated Effective Mass (IEM) The concept of Integrated Effective Mass (IEM) quantifies the inertia of the manipulator in the direction of contact. It is derived from the dynamic parameters of the manipulator using the Jacobian matrix J, which maps the joint velocities to the end-effector velocities. The IEM is derived from the effective mass of the system. The effective mass is the apparent inertia of a robotic mani-pulator (or object) in a particular direction of motion or force, and it quantifies the resistance offered by the end effector of a robot when a force is applied in a given direction. Mathematically, this is represented as where: J = Jacobian matrix (linear velocity part); M = inertia matrix in joint space; n = unit vector along the contact direction; The integrated effective mass is a cumulative or averaged measure of the effective mass across a range of configurations, or along a manipulator path. It captures the overall dynamic behavior of the robot during a capture or contact maneuver involving multiple joints and motion segments. The total mass of the system can be expressed using the integrated effective mass of a continuous object (robotic manipulator) as follows: where - the unit norm direction vector; - the vector from mass center of target to contact point; - effective mass of the robotic manipulator and mass of mass of the target respectively. The effective mass of the robotic manipulator depicts the total mass as a continuous object and is determined as follows [60; 61]: where is a unit direction vector, is called the effective mass, and with the Jacobian matrix corresponding to the linear velocity. 2.5. Jacobian Modeling and DH Parameters Before determining the effective mass of the end effector, the Jacobian matrix for the robotic manipulator must be determined. For the 7-DOF free-floating space robot (modeled after the KUKA system), the Jacobian is derived using Denavit-Hartenberg parameters. This mathematical framework allows the mapping of joint-space motion to the robot’s operational space [62]. A symbolic, compact form of the Jacobian matrix was used to compute the dynamic behavior and ultimately evaluate the effective mass across different configurations [63]. The D-H parameters for the KUKA robot shown in Figure 3 are listed in Table 2. Figure 3. The 2D orientation of 7-DOF free-floating space robot capturing tumbling target S o u r c e: by L. Zhang [54] Table 2 D-H Parameter of seven DOF KUKA robot Link ai-1(m) αi-1(0) di(m) θ(0) 1 0 0 0.6 θ 1 2 0 90 0 θ 2 3 0 -90 0 θ 3 4 1 0 0 θ 4 5 1 0 0.2 θ 5 6 0 90 0.2 θ 6 7 0 -90 0 θ 7 S o u r c e: by J. Chimento [62] The Jacobian matrix below has been compacted by symbolizing only entries with more than two mathematical operations. Trigonometric shorthand was used as follows: s_456 = sin(theta_4+theta_5+theta_6). Full Jacobean Matrix ; ; ; The effective mass of the end-effector is derived from the Jacobian matrix of the system using the dynamic parameters of the space robot listed in Table 3. The corresponding simulation results using MATLAB for the effective mass are presented in Table 4. Table 3 Dynamic Parameter of the Robot Part mi ,kg Ii ,kg m2 Link 1 5 diag([0.01, 0.02, 0.02]) Link 2 5 diag([0.02, 0.01, 0.02]) Link 3 10 diag([0.84, 0.01, 0.84]) Link 4 10 diag([0.01, 0.84, 0.84]) Link 5 5 diag([0.02, 0.02, 0.01]) Link 6 5 diag([0.02, 0.02, 0.01]) Link 7 8 diag([0.03, 0.03, 0.01]) Base 1,000 diag([500, 500, 500]) Target 200 diag([100, 100, 100]) S o u r c e: by L. Skrinjar, J. Slaviˇc, and M. Bolteˇzar [55] Table 4 Effective Mass of the End Effector No Configuration Effective Mass, kg 1 30, -55, 60, -70, 35, 20, -40 1.7377 2 -20, 85, -110, 30, -45, 60, 150 0.0413 3 90, -75, 40, -130, 70, 15, -20 1.6088 4 -60, 120, -35, 55, 100, 80, 25 0.7090 5 10, -140, 95, -15, 60, -10, 130 0.1348 6 -85, 40, -70, 20, -11, 120, 55 0.2359 7 135, -50, 75, -90, 15, 100, -35 0.0841 8 -30, 110, -100, 70, 40, -20, 60 0.1355 9 70, -100, 55, -45, 140, -60, 5 0.2162 10 -150, 25, -20, 115, -95, 45, -70 0.4346 S o u r c e: by S. Doliwa [63] 2.6. Singularity Analysis Singularity analysis plays a critical role in en-suring that a free-flying space robot never reaches a state of uncontrollability, where motion along one or more Cartesian directions becomes impossible or demands unrealistically large joint velocities. A singularity occurs when the manipulator’s Jacobian matrix J(θ) loses rank, thereby reducing its ability to map joint velocities into end-effector velocities [64]. For a redundant 7-DOF manipulator, such as the KUKA LBR considered in this study, singularities may appear in specific configurations where the rank of J drops below six. This condition typically arises due to certain joint alignments, which lead to the loss of motion capability in one or more directions and, consequently, the risk of control instability [65; 66]. In this study, singularity tests were performed across ten representative configurations, as shown in Table 4, using a MATLAB simulation. The results of the evaluation on singularity are presented in Table 5 below. In every case, the Jacobian maintained full rank (Rank (J) = 6), indicating that none of the chosen poses were singular. Table 5 Singularity of 7-DOF Space Robots Configuration Rank Singularity 1 Rank(J) = 6 Not Singular 2 Rank(J) = 6 Not Singular 3 Rank(J) = 6 Not Singular 4 Rank(J) = 6 Not Singular 5 Rank(J) = 6 Not Singular 6 Rank(J) = 6 Not Singular 7 Rank(J) = 6 Not Singular 8 Rank(J) = 6 Not Singular 9 Rank(J) = 6 Not Singular 10 Rank(J) = 6 Not Singular S o u r c e: by A. Mueller [66] However, points along a trajectory may approach near-singular conditions, underscoring the importance of careful motion planning. Entering a singular configuration during microgravity ope-rations can significantly amplify manipulator - base coupling effects, driving joint torques beyond their limits and potentially destabilizing the spacecraft attitude [68]. Once the singularity of the workspace has been identified, IEM of the space robot manipulator at a given configuration can be easily determined using the equation. Therefore, singularity analysis should not be treated as a routine mathematical exercise but as a vital safety measure for ensuring robust, stable, and precise capture maneuvers in free-flying space robotics. 3. Results and Discussion 3.1. Numerical Simulation of a 7-DOF Free-Floating Space Robot A 7-DOF free-floating space robot with a = 0.6 m, b = 0.2 m, c = 0.2 m, d = 1.0 m, e = 1.0 m, f = 0.2 m, g = 0.2 m, h = 0.2 m and k = 0.6 m. and are both expressed in the target coordinate frame [54]. The unit vector according to the calculation is expressed as follows: . The dynamics parameters of the space robot and target are listed in Table 1. Table 6 was derived from the Mie equation above and computed using MATLAB. It summarizes the simulated integrated effective mass (Mie) for each configuration. It also lists the simulated maximum contact force during on-orbit servicing between the space robot manipulator and the target, which was obtained using the Hunt - Crossley model with a hysteresis damping factor. From Table 6, we can see that the maximum force has different values for different integrated effective masses, which are the results obtained by randomly varying the configuration ten times. The reduced maximum contact force created between the robotic manipulator and the tumbling target was observed in configuration 2. Figure 4 was plotted using MATLAB from the data in Table 6 and reveals that as the integrated effective mass increases, the contact force during space robot captures increases nonlinearly, particularly beyond a certain threshold. This indicates a higher momentum transfer and impact severity. Minimizing IEM is essential for reducing capture forces and ensuring safer and more stable robotic interactions during satellite servicing and debris capture missions. Table 6 Integrated Effective Mass for Different Configuration NO Configuration Integrated Effective Mass, kg Maximum Force, N 1 30, -55, 60, -70, 35, 20, -40 1.7199 4.1421e + 03 N 2 -20, 85, -110, 30, -45, 60, 150 0.0413 4.4207e + 02 N 3 90, -75, 40, -130, 70, 15, -20 1.5935 3.9567e + 03 N 4 -60, 120, -35, 55, 100, 80, 25 0.7060 2.4278e + 03 N 5 10, -140, 95, -15, 60, -10, 130 0.1347 8.9855e + 02 N 6 -85, 40, -70, 20, -11, 120, 55 0.2355 1.2564e + 03 N 7 120, -40, 60, -70, 35, 60, -20 0.0841 6.7733e + 02 N 8 -30, 110, -100, 70, 40, -20, 60 0.1354 9.0135e + 02 N 9 70, -100, 55, -45, 140, -60, 5 0.2159 1.1925e + 03 N 10 -150, 25, -20, 115, -95, 45, -70 0.4334 1.8116e + 03 N S o u r c e: by L. Zhang [54] Figure 4. Effect of Integrated Effective Mass to Reduce Contact Force S o u r c e: by P. Flores and H.M. Lankarani [10] Figure 5. Comparison of Hysteresis Damping factor for different models S o u r c e: by P. Flores and H.M. Lankarani [10] Figure 6. Effects of Coefficient of Restitution the Maximum Force And Deformation S o u r c e: by P. Flores and H.M. Lankarani [10] As demonstrated in the above figure (Figure 5), the contact models of Lee - Wang, Lankarani - Nikravesh, Herbert - McWhannell, and Hunt - Crossley exhibit reduced dissipation energy owing to their diminished hysteresis damping factor. However, from the figure, we can see that the Lee-Wang, Lankarani - Nikravesh, and Herbert - McWhannel models have higher deformation than the Hunt - Crossley model. For the above reasons, the Hunt - Crossley contact force model was selected to model the contact forces between the robotic manipulator and the tumbling target. When the coefficient of restitution was reduced, the maximum force and maximum deformation were reduced; however, this resulted in an increase in the contact duration, which resulted in energy dissipation, as shown in Figure 6. Therefore, it is necessary to use a coefficient of restitution of approximately 0.8. Figure 7 shows the simulation performed in MATLAB to compare the theoretical data with the numerical data to verify the results in depicting the error propagation. Consequently, we can observe from the figure that the error between the maximum theoretical and numeric forces was significantly reduced, as shown in Figure 7. Figure 7. Reduced Error S o u r c e: by P. Flores and H.M. Lankarani [10] Conclusion This study demonstrated how the configuration of a free-floating 7-DOF space robot strongly in-fluences the capture dynamics during satellite ser-vicing and debris removal. By analyzing the integrated effective mass (IEM) across different joint configurations, it was observed that the con-tact forces increased nonlinearly with the IEM. Capture forces were dramatically reduced by nearly an order of magnitude in low-IEM configurations compared to higher IEM poses. For instance, an IEM of 0.0413 kg produced only 442 N, whereas an IEM of 1.7199 kg generated more than 4142 N. The Hunt - Crossley model was effective in re-presenting the contact dynamics, and a restitution coefficient of approximately 0.8 provided a good balance between reducing the rebound and main-taining the capture reliability. These results confirm that configuration optimization guided by IEM analysis is a powerful strategy for ensuring safe, more stable, and reliable space robot operations. However, this study assumes rigid targets and ideal conditions, overlooking flexible structures, multi-contact, and sensor noise. Future efforts should integrate trajectory planning, flexible modeling, and adaptive learning-based control to address uncertainties, paving the way for more robust, efficient, and sustainable space robotics missions.

About the authors

Yeshurun A. Adde

Addis Ababa University

Author for correspondence.
Email: kibret10@gmail.com
ORCID iD: 0000-0001-5137-0667

Doctor of Philosophy (Physics), PhD Scholar, School of Mechanical & Industrial Engineering, College of Technology & Built Environment-AAiT

Addis Ababa, Ethiopia

Yury N. Razoumny

RUDN University

Email: yury.razoumny@gmail.com
ORCID iD: 0000-0003-1337-5672
SPIN-code: 7704-4720

Doctor of Sciences (Techn.), Director of the Academy of Engineering, Head of the Department of Mechanics and Control Processes, Academy of Engineering

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Araya Abera Betelie

Addis Ababa University

Email: arsame2008@gmail.com
ORCID iD: 0009-0008-8761-5644

Doctor of Philosophy (Mech.), Assistant Professor, School of Mechanical & Industrial Engineering, College of Technology & Built Environment-AAiT

Addis Ababa, Ethiopia

Biruk Degefu

Addis Ababa University

Email: birukdegefu16@gmail.com
ORCID iD: 0009-0009-2368-7371

Bachelor of Science (Mech.), MSc Student, School of Mechanical & Industrial Engineering, College of Technology & Built Environment-AAiT

Addis Ababa, Ethiopia

References

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